Nonhomogeneous dual wavelet frames and mixed oblique extension principles in Sobolev spaces

References

• Christensen O . An introduction to frames and Riesz bases. Boston (MA): Birkhäuser; 2003.
   Google Scholar
• Duffin RJ , Schaeffer AC . A class of nonharmonic Fourier series. Trans Am Math Soc. 1952;72:341–366.
   Web of Science ®Google Scholar
• Hernández E , Weiss G . A first course on wavelets. Boca Raton (FL): CRC Press; 1996.
   Google Scholar
• Young RM . An introduction to nonharmonic Fourier series. New York (NY): Academic Press; 1980.
   Google Scholar
• Benedetto JJ , Li S . The theory of multiresolution analysis frames and applications to filter banks. Appl Comput Harmon Anal. 1998;5:389–427.
   Web of Science ®Google Scholar
• Bownik M . Tight frames of multidimensional wavelets. J Fourier Anal Appl. 1997;3:525–542.
   Web of Science ®Google Scholar
• Calogero A . A characterization of wavelets on general lattices. J Geom Anal. 2000;10:597–622.
   Google Scholar
• Chui CK , Shi X , Stöckler J . Affine frames, quasi-affine frames, and their duals. Adv Comput Math. 1998;8:1–17.
   Web of Science ®Google Scholar
• Chui CK , He W . Compactly supported tight frames associated with refinable functions. Appl Comput Harmon Anal. 2000;8:293–319.
   Web of Science ®Google Scholar
• Chui CK , He W , Stöckler J . Compactly supported tight and sibling frames with maximum vanishing moments. Appl Comput Harmon Anal. 2002;13:224–262.
   Web of Science ®Google Scholar
• Gripenberg G . A necessary and sufficient condition for the existence of a father wavelet. Studia Math. 1995;114:207–226.
   Web of Science ®Google Scholar
• Han B . On dual wavelet tight frames. Appl Comput Harmon Anal. 1997;4:380–413.
   Web of Science ®Google Scholar
• Gröchenig K , Ron A . Tight compactly supported wavelet frames of arbitrarily high smoothness. Proc Am Math Soc. 1998;126:1101–1107.
   Web of Science ®Google Scholar
• Lawton W . Tight frames of compactly supported affine wavelets. J Math Phys. 1990;31:1898–1901.
   Web of Science ®Google Scholar
• Ron A , Shen Z . Affine systems in L2(ℝ2): the analysis of the analysis operator. J Funct Anal. 1997;148:408–447.
   Web of Science ®Google Scholar
• Ron A , Shen Z . Affine systems in L2(ℝ2) II: dual systems. J Fourier Anal Appl. 1997;3:617–637.
   Web of Science ®Google Scholar
• Bownik M . A characterization of affine dual frames in L2(ℝn). Appl Comput Harmon Anal. 2000;8:203–221.
   Web of Science ®Google Scholar
• Christensen O , Kim HO , Kim RY . On extensions of wavelet systems to dual pairs of frames. Adv Comput Math. 2016;42:489–503.
   Web of Science ®Google Scholar
• Christensen O , Kim HO , Kim RY . On Parseval wavelet frames with two or three generators via the unitary extension principle. Can Math Bull. 2014;57:254–263.
   Web of Science ®Google Scholar
• Li Y-Z , Zhou F-Y . Affine and quasi-affine dual wavelet frames in reducing subspaces of L2(ℝ2). Acta Math Sinica (Chin Ser.). 2010;53:551–562.
   Google Scholar
• Daubechies I , Han B , Ron A , et al . MRA-based constructions of wavelet frames. Appl Comput Harmon Anal. 2003;14:1–46.
   Web of Science ®Google Scholar
• Petukhov AP . Explicit construction of framelets. Appl Comput Harmon Anal. 2001;11:313–327.
   Web of Science ®Google Scholar
• Atreas N , Melas A , Stavropoulos T . Affine dual frames and extension principles. Appl Comput Harmon Anal. 2014;36:51–62.
   Web of Science ®Google Scholar
• Bownik M , Rzeszotnik Z . Construction and reconstruction of tight framelets and wavelets via matrix mask functions. J Funct Anal. 2009;256:1065–1105.
   Web of Science ®Google Scholar
• Daubechies I , Han B . Pairs of dual wavelet frames from any two refinable functions. Constr Approx. 2004;20:325–352.
   Web of Science ®Google Scholar
• Ehler M . On multivariate compactly supported bi-frames. J Fourier Anal Appl. 2007;13:511–532.
   Web of Science ®Google Scholar
• Ehler M , Han B . Wavelet bi-frames with few generators from multivariate refinable functions. Appl Comput Harmon Anal. 2008;25:407–414.
   Web of Science ®Google Scholar
• Jia H-F , Li Y-Z . Refinable function-based construction of weak (quasi-)affine bi-frames. J Fourier Anal Appl. 2014;20:1145–1170.
   Web of Science ®Google Scholar
• Li Y-Z , Zhou F-Y . GMRA-based construction of framelets in reducing subspaces of L2(ℝ2). Int J Wavelets Multiresolut Inf Process. 2011;9:237–268.
   Web of Science ®Google Scholar
• Lian Q-F , Li Y-Z . Reducing subspace frame multiresolution analysis and frame wavelets. Commun Pure Appl Anal. 2007;6:741–756.
   Web of Science ®Google Scholar
• Stavropoulos T . The geometry of extension principles. Houston J Math. 2012;38:833–853.
   Web of Science ®Google Scholar
• Zhou F-Y , Li Y-Z . Multivariate FMRAs and FMRA frame wavelets for reducing subspaces of L2(ℝ2). Kyoto J Math. 2010;50:83–99.
   Web of Science ®Google Scholar
• Han B . Nonhomogeneous wavelet systems in high dimensions. Appl Comput Harmon Anal. 2012;32:169–196.
   Web of Science ®Google Scholar
• Han B . Wavelets and framelets within the framework of nonhomogeneous wavelet systems. In: Neamtu M , Schumaker LL , editors. Approximation theory XIII: San Antonio 2010. New York (NY): Springer; 2012. p. 121–161. (Springer proceedings in mathematics; vol. 13).
   Google Scholar
• Han B . Pairs of frequency-based nonhomogeneous dual wavelet frames in the distribution space. Appl Comput Harmon Anal. 2010;29:330–353.
   Web of Science ®Google Scholar
• Chui CK , Sun Q . Affine frame decompositions and shift-invariant spaces. Appl Comput Harmon Anal. 2006;20:74–107.
   Web of Science ®Google Scholar
• Ehler M . The multiresolution structure of pairs of dual wavelet frames for a pair of Sobolev spaces. Jaen J Approx. 2010;2:193–214.
   Google Scholar
• Ehler M . Nonlinear approximation schemes associated with nonseparable wavelet bi-frames. J Approx Theory. 2009;161:292–313.
   Web of Science ®Google Scholar
• Han B , Shen Z . Dual wavelet frames and Riesz bases in Sobolev spaces. Constr Approx. 2009;29:369–406.
   Web of Science ®Google Scholar
• Han B , Shen Z . Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames. Israel J Math. 2009;172:371–398.
   Web of Science ®Google Scholar
• Jia RQ , Wang JZ , Zhou DX . Compactly supported wavelet bases for Sobolev spaces. Appl Comput Harmon Anal. 2003;15:224–241.
   Web of Science ®Google Scholar
• Li Y , Yang S . Multiwavelet sampling theorem in Sobolev spaces. Sci China Math. 2010;53:3197–3214.
   Web of Science ®Google Scholar
• Romero JR , Alexander SK , Baid S , et al . The geometry and the analytic properties of isotropic multiresolution analysis. Adv Comput Math. 2009;31:283–328.
   Web of Science ®Google Scholar
• Borup L , Gribonval R , Nielsen M . Tight wavelet frames in Lebesgue and Sobolev spaces. J Funct Spaces Appl. 2004;2:227–252.
   Google Scholar
• Borup L , Gribonval R , Nielsen M . Bi-framelet systems with few vanishing moments characterize Besov spaces. Appl Comput Harmon Anal. 2004;17:3–28.
   Web of Science ®Google Scholar
• Li Y , Yang S , Yuan D . Bessel multiwavelet sequences and dual multiframelets in Sobolev spaces. Adv Comput Math. 2013;38:491–529.
   Web of Science ®Google Scholar